James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

792 Chapter 12 Vectors and the Geometry of Space
x
z
O
FIGURE 1
Coordinate axes
x
z
FIGURE 2
Right-hand rule
y
y
3D Space
To locate a point in a plane, we need two numbers. We know that any point in the plane
can be represented as an ordered pair sa, bd of real numbers, where a is the x-coordinate
and b is the y-coordinate. For this reason, a plane is called two-dimensional. To locate a
point in space, three numbers are required. We represent any point in space by an ordered
triple sa, b, cd of real numbers.
In order to represent points in space, we first choose a fixed point O (the origin) and
three directed lines through O that are perpendicular to each other, called the coordinate
axes and labeled the x-axis, y-axis, and z-axis. Usually we think of the x- and y-axes as
being horizontal and the z-axis as being vertical, and we draw the orientation of the axes
as in Figure 1. The direction of the z-axis is determined by the right-hand rule as illustrated
in Figure 2: If you curl the fingers of your right hand around the z-axis in the direction
of a 90° counterclockwise rotation from the positive x-axis to the positive y-axis,
then your thumb points in the positive direction of the z-axis.
The three coordinate axes determine the three coordinate planes illustrated in Figure
3(a). The xy-plane is the plane that contains the x- and y-axes; the yz-plane contains
the y- and z-axes; the xz-plane contains the x- and z-axes. These three coordinate planes
divide space into eight parts, called octants. The first octant, in the foreground, is determined
by the positive axes.
z
z
x
xz-plane
O
xy-plane
yz-plane
y
x
left wall
O
floor
right wall
y
FIGURE 3
(a) Coordinate planes
(b)
x
FIGURE 4
a
z
O
b
P(a, b, c)
c
y
Because many people have some difficulty visualizing diagrams of three-dimensional
figures, you may find it helpful to do the following [see Figure 3(b)]. Look at any bottom
corner of a room and call the corner the origin. The wall on your left is in the xz-plane,
the wall on your right is in the yz-plane, and the floor is in the xy-plane. The x-axis runs
along the intersection of the floor and the left wall. The y-axis runs along the intersection
of the floor and the right wall. The z-axis runs up from the floor toward the ceiling along
the intersection of the two walls. You are situated in the first octant, and you can now
imagine seven other rooms situated in the other seven octants (three on the same floor
and four on the floor below), all connected by the common corner point O.
Now if P is any point in space, let a be the (directed) distance from the yz-plane to P,
let b be the distance from the xz-plane to P, and let c be the distance from the xy-plane to
P. We represent the point P by the ordered triple sa, b, cd of real numbers and we call
a, b, and c the coordinates of P; a is the x-coordinate, b is the y-coordinate, and c is the
z-coordinate. Thus, to locate the point sa, b, cd, we can start at the origin O and move
a units along the x-axis, then b units parallel to the y-axis, and then c units parallel to the
z-axis as in Figure 4.
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